Existence and multiplicity of positive solutions for the p-Laplacian with nonlocal coefficient.
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Date
2008
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Abstract
We consider the Dirichlet problem with nonlocal coefficient given by −a(Ω|u|q dx)_pu = w(x)f (u) in a bounded, smooth domain Ω ⊂ Rn (n _ 2), where _p is the p-Laplacian, w is a weight function and the nonlinearity f (u) satisfies certain local bounds. In contrast with the hypotheses usually made, no asymptotic behavior is assumed on f . We assume that the nonlocal coefficient a(_Ω|u|q dx) (q _ 1) is defined by a continuous and nondecreasing function a : [0,∞)→[0,∞) satisfying a(t) > 0 for t > 0 and a(0) _ 0. A positive solution is obtained by applying the Schauder Fixed Point Theorem. The case a(t) = tγ/q (0 < γ < p − 1) will be considered as an example where asymptotic conditions on the nonlinearity provide the existence of a sequence of positive solutions for the problem with arbitrarily large sup norm.
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Keywords
Laplacian, Nonlocal coefficient, Existence and multiplicity of positive solutions
Citation
BUENO, H. et al. Existence and multiplicity of positive solutions for the p-Laplacian with nonlocal coefficient. Journal of Mathematical Analysis and Applications, v. 343, p. 151-158, 2008. Disponível em: <http://www.sciencedirect.com/science/article/pii/S0022247X08000036>. Acesso em: 10 mar. 2015.